If a magnetic field is considered only in the direction along the \(z\) -axis, as is the case of Sec. , the force (the third constraint in (REF )) can be zeroed a priori before solving the optimization problem. This is done by using the null space of a matrix, the rows of which are specific components of a magnetic field operator and of its first derivatives, as a solution subspace. Namely, components \({B}_x,{B}_y\)  and \({B}_{z,i}, \ \forall i \in \lbrace x,y,z\rbrace \)  are used in Sec. . Consulting the law of Biot and Savart (REF ), members of this subspace are invariant with respect to the \(\sigma _z C_{4z}\)  operation [1]}, whenever the underlying geometry allows it. This is advantageously used in topology optimization, where the constraint on zero force in the trapping center is enforced by aforementioned symmetry of structure and feeding, greatly reducing the optimization complexity.
